完全图的Cantesian积的L(2,1)-圆标定

For positive integers j and k with j ≥ k, an L（j, k）-labeling of a graph G is an assignment of nonnegative integers to V（G） such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L（j, k）-labeling of a graph G is the difference between the maximum and minimum integers it uses. The λj, k-number of G is the minimum span taken over all L（j, k）-labelings of G. An m-（j, k）-circular labeling of a graph G is a function f ： V（G） →{0, 1, 2,..., m - 1} such that ｜f（u） - f（v）｜m ≥ j if u and v are adjacent; and ｜f（u） - f（v）｜m 〉 k ifu and v are at distance two, where ｜x｜m = min{｜xl｜, m-｜x｜}. The minimum integer m such that there exists an m-（j, k）-circular labeling of G is called the σj,k-number of G and is denoted by σj,k（G）. This paper determines the σ2,1-number of the Cartesian product of any three complete graphs.

L(2,1)-Circular Labelings of Cartesian Products of Complete Graphs

For positive integers $j$ and $k$ with $j\geq k$, an $L(j,k)$-labeling of a graph $G$ is an assignment of nonnegative integers to $V(G)$ such that the difference between labels of adjacent vertices is at least $j$, and the difference between labels of vertices that are distance two apart is at least $k$. The span of an $L(j,k)$-labeling of a graph $G$ is the difference between the maximum and minimum integers it uses. The $\lambda_{j,k}$-number of $G$ is the minimum span taken over all $L(j,k)$-labelings of $G$. An $m$-$(j,k)$-circular labeling of a graph $G$ is a function $f: V(G)\rightarrow \{0,1,2,\ldots,m-1\}$ such that $|f(u)-f(v)|_{m}\geq j$ if $u$ and $v$ are adjacent; and $|f(u)-f(v)|_{m}\geq k$ if $u$ and $v$ are at distance two, where $|x|_{m}=\min\{|x|,m-|x|\}$. The minimum integer $m$ such that there exists an $m$-$(j,k)$-circular labeling of $G$ is called the $\sigma_{j,k}$-number of $G$ and is denoted by $\sigma_{j,k}(G)$. This paper determines the $\sigma_{2,1}$-number of the Cartesian product of any three complete graphs.